A tractable family of remainder- form mixed-monotone decomposition functions was proposed in this paper for a relatively large class of nonsmooth systems called either-sided locally Lipschitz systems that is proven to include locally Lipschitz continuous systems.
We also characterized the lower and upper bounds for the over-approximation errors when using the proposed remainder-form decomposition functions to over-approximate the true range/image set of a nonsmooth, nonlinear mapping, where the lower bound is achieved by our proposed tight and tractable decomposition function, which was further proven to be tighter than the one introduced in [1]}.
 In addition, a novel set inversion algorithm based on decomposition functions was developed to further refine/update the reachabl e sets when knowledge of state constraints and/or when measurements are available, which can be applied for constrained reachability analysis and interval observer design for bounded-error continuous-time, discrete-time or sampled-data systems. Finally, the effectiveness of our proposed mixed-monotone decomposition functions is demonstrated using several benchmark examples in the literature.
